# Can magnetic field change kinetic energy?

An electron and a proton are moving under the influence of mutual forces. In calculating the change in the kinetic energy of the system during motion, one ignores the magnetic force of one on another. This is, because

(a) the two magnetic forces are equal and opposite, so they produce no net effect

(b) the magnetic forces do not work on each particle

(c) the magnetic forces do equal and opposite (but non-zero) work on each particle

(d) the magnetic forces are necessarily negligible

Magnetic field does no work. The Lorentz force due to a point charge is $$\vec{F} = q(\vec{E}+\vec{v}\times\vec{B}).$$ The force due to the magnetic field is $$\vec{F}_{mag} = q(\vec{v}\times\vec{B}) .$$ The work done on $$q$$ due to the magnetic force per unit time is $$P_{mag} = \vec{F}_{mag}·\vec{v} = q(\vec{v}\times\vec{B})·\vec{v} = q(\vec{v}\times\vec{v})·\vec{B} = 0.$$ Thus, quite generally, forces due to magnetic fields do no work, because $$\vec{F}_{mag}$$ is orthogonal to $$\vec{v}$$.